WASHINGTON — President-elect Donald J. Trump is making his influence felt on government funding deliberations in Congress, pressing Republican leaders to postpone action on spending bills until after he takes office. Speaker Paul D. Ryan said on Thursday that the House would go along with the incoming administration’s request and pass a stopgap spending bill that would keep the government running at current spending levels until March. “I think they would like to have a say-so on how money’s going to be spent going into the next year,” he told reporters. But Republican leaders in the Senate did not immediately sign on to the plan, reflecting their desire to get contentious spending battles out of the way this year. Congress had been working toward agreement on spending bills by Dec. 9, when the current funding agreement expires. Senator Mitch McConnell of Kentucky, the Republican leader, left open the possibility of an agreement with the House. “Discussions are also ongoing about how to fund the government and for how long,” he said on the floor on Thursday. But signaling that the longstanding tensions between the House and Senate may not abate under a more unified Republican government, some Senate Republicans expressed concern that waiting until early next year could distract them from other legislative priorities. “My opinion is that it would be better to get this year done now,” said Senator Roy Blunt, Republican of Missouri and a member of the Appropriations Committee. “My preference is to do it now and not get bogged down early next year.” The issue flared as both parties on Capitol Hill continued to adapt to Mr. Trump’s victory and his promise to shake up the way Washington works. Vice President-elect Mike Pence visited the Capitol to meet with Republican and Democratic leaders, as well as House Republicans. And the House Democratic leader, Nancy Pelosi of California, drew a challenger on Thursday in her campaign to retain her post, as Representative Tim Ryan of Ohio said the election outcome required Democrats to take a new approach. Startled by the number of working-class voters in states like Michigan, Ohio and Pennsylvania that shifted their support to Republicans — particularly Mr. Trump —Democrats have spent the week contemplating how to recapture those voters. Mr. Ryan, 43, who represents the quintessentially Rust Belt area of Youngstown, Ohio, said he believed he could help Democrats reach them.